\documentclass{article}
\usepackage{epsfig}
\pagestyle{empty}
\begin{document}
\[ L(\mathbf{q},\mathbf{\dot{q}})=T(\mathbf{q},\mathbf{qp})+V(\mathbf{q}) \]
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$ V(\mathbf{q}) $
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$ T(\mathbf{q},\mathbf{qp}})$
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\[ T(\mathbf{q},\mathbf{qp})=\frac{1}{2}\sum_{at} m_{at}\left(\sum_{i} \frac{\partial \vec{R}_{at}}{\partial q_i}qp_i\right)^2 \\ =\frac{1}{2}\mathbf{qp^T}\mathbf{\underline{A}} \left(\mathbf{q}\right)\mathbf{qp} \]
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$ \underline{A} $
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\[ A_{ij}=\frac{1}{2}\sum_{at} m_{at} \frac{\partial \vec{R}_{at}}{\partial q_i}. \frac{\partial \vec{R}_{at}}{\partial q_j} \]
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\[ Q(T)=\frac{1}{\delta}\left(2\pi k_BT\right)^{n/2} \int |\underline{A}(\mathbf{q})|^{1/2} \exp\left(-V(\mathbf{q})/k_B T\right)d\mathbf{q} \]
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$ k_B $
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$ T $
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$ n $
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$ \mathbf{q} $
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$ \delta $
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\[ g(E)=\frac{1}{\delta \left(N_{av}\right)^{n/2}\Gamma(n/2) } \left(\frac{2\pi}{h^2}\right)^{n/2} \int |\underline{A}(\mathbf{q})|^{1/2}\left(E-V(\mathbf{q})\right)^{n/2-1}d\mathbf{q} \]
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$ N_{av} $
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$ h $
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$ \Gamma $
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$ E $
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$ V $
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\[ S(T)=R \left( ln(Q) + T \left( \frac{\partial ln(Q)}{dT} \right)_{V} \right) \]
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\[ E_{Therm}(T) = RT^2 \left( \frac{\partial ln(Q)}{dT} \right)_{V} \]
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\[ msv(T) = RT^2 \left( \frac{\partial E_{Therm}}{dT} \right)_{V} \]
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\[ \[ \frac{d}{dt}\frac{\partial L}{\partial{qp}_i} = \frac{\partial L}{\partial q_i} \] \]
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\[ \[ \sum_j\frac{\partial^2 L}{\partial {qp}_i\partial{qp}_j} \qpp_j+\sum_j\frac{\partial^2 L}{\partial {qp}_i\partial q_j} {qp}_j=\frac{\partial L}{\partial q_i}. \] \]
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$ L $
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$ i $
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$ A.qpp+B.qp=v $
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\[ \[ A_{ij}=\frac{\partial^2 L}{\partial {qp}_i\partial {qp}_j},\ B_{ij}=\frac{\partial^2 L}{\partial {qp}_i\partial q_j},\ v_i=\frac{\partial L}{\partial q_i}. \] \]
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$ j $
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$ q_j $
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$ qp_j $
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$ qpp_j $
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$ qpp_i $
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$ 2J+1 $
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\[ E_j=B*j*(j+1) \]
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$ B $
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\[ E(\mathbf{n})= G_0 + \omega(n_*+\frac{1}{2}) + \sum_{i} \nu_{*i}(n_*+\frac{1}{2})(n_i+\frac{1}{2}) + \sum_{ij} \nu_{*ij}(n_*+\frac{1}{2})(n_i+\frac{1}{2})(n_j+\frac{1}{2}) + ... \]
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$ G_0 $
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$ \omega_* $
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$ \nu_{*j} $
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$ \mathbf{n} $
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\[ E(\mathbf{n})= G_0 + \omega(n_*+\frac{1}{2}) \]
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\[ E(\mathbf{n})= G_0 + \omega(n_*+\frac{1}{2}) + \sum_{i} \nu_{*i}(n_*+\frac{1}{2})(n_i+\frac{1}{2}) \]
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\[ K_{eff}(T)=\int K(q)P(q,T)dq \]
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\[ P(\theta,T)=\frac{\exp\left(-V(q)/RT\right)}{\int \exp\left(-V(q)/RT\right)dq} \]
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$ q $
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$ \varepsilon_i^{eff}(T) $
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\[ H^{1D}_{eff}=-\frac{\hbar^2}{2K_{eff}}\frac{\partial^2}{\partial q^2}+V(q) \]
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\[ Q^{1D}_{eff}(T)=\frac{1}{\sigma} \sum_i \text{exp}\left( -\varepsilon_i^{eff}(T)/k_BT \right) \]
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$ J $
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$ I_i $
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\[ q_r=(\pi.I)^{0.5}\left((8\pi^2k_bT)/(h*h)\right)^{1.5}/\sigma \]
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$ I= \Pi_i I_i $
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$ k_b $
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$ \sigma $
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$ CH_3 $
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$ i_0 $
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$ q_0 $
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$ i_1 $
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$ q_1 $
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$ i_2 $
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$ q_2 $
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$ i_3 $
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$ q_3 $
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$ i_4 $
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$ q_4 $
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$ i_5 $
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$ q_5 $
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$ d\mathbf{q} $
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\end{document}
